Family of Quadratic Functions
Lesson 5.5a
General Form
• Quadratic functions have the standard
form
y = ax2 + bx + c
 a, b, and c are constants
 a≠0
(why?)
• Quadratic functions graph as a parabola
Zeros of the Quadratic
• Zeros are where the function crosses the xaxis
 Where y = 0
• Consider possible numbers of zeros

None (or two complex)
One
Two
Axis of Symmetry
• Parabolas are symmetric
about a vertical axis
• For y = ax2 + bx + c the axis
of symmetry is at
b
x
2a
• Given y = 3x2 + 8x
 What is the axis of symmetry?
Vertex of the Parabola
• The vertex is the “point” of the
parabola
 The minimum value
 Can also be a maximum
• What is the x-value of the
vertex? x  b
2a
• How can we find the
y-value?
 b 
y  f ( x)  f  
 2a 
Vertex of the Parabola
• Given f(x) = x2 + 2x – 8
• What is the x-value of the vertex?
b 2
x

 1
2a 2 1
• What is the y-value of the vertex?
f (1)  1  2  9  9
• The vertex is at (-1, -9)
Vertex of the Parabola
• Given f(x) = x2 + 2x – 8
 Graph shows vertex at (-1, -9)
• Note calculator’s ability to find vertex
(minimum or maximum)
Shifting and Stretching
• Start with f(x) = x2
• Determine the results of transformations
 ___ f(x + a) = x2 + 2ax + a2
 ___ f(x) + a = x2 + a
 ___ a * f(x) = ax2
 ___ f(a*x) = a2x2
a) horizontal shift
b) vertical stretch or
squeeze
c) horizontal stretch or
squeeze
d) vertical shift
e) none of these
Other Quadratic Forms
• Standard form
y = ax2 + bx + c
• Vertex form
y = a (x – h)2 + k
Experiment with
Quadratic Function
Spreadsheet
 Then (h,k) is the vertex
• Given f(x) = x2 + 2x – 8
 Change to vertex form
 Hint, use completing the square
Vertex Form
• Changing to vertex form
Add something in to
make a perfect
square trinomial
y  x2  2x  8
y  x  2x 
8
2

y  x

2

Now create a
binomial squared
Subtract the same
amount to keep it even.
This gives us the
ordered pair (h,k)
Assignment
• Lesson 5.5a
• Page 231
• Exercises 1 – 25 odd